Composition and inversion in the <Affine group of the complex line> are
$$
f_{a,b}\circ f_{a',b'}=f_{aa',\,ab'+b},
\qquad
f_{a,b}^{-1}=f_{a^{-1},\,-a^{-1}b}.
$$
The map
$$
\rho:G\longrightarrow\mathbb C^\times,\qquad
\rho(f_{a,b})=a
$$
is therefore a surjective <group homomorphism> with kernel
$$
N=\{f_{1,b}:b\in\mathbb C\}\cong(\mathbb C,+).
$$
Its target is <Abelian>, so $[G,G]\subseteq N$. On the other hand, the stated computation gives
$$
[f_{a,1},f_{1,b}]=f_{1,b(1-a^{-1})}.
$$
Fixing any $a\ne1$ and varying $b$ produces every translation. Thus $N\subseteq[G,G]$, and hence
$$
\boxed{[G,G]=N\cong(\mathbb C,+),\qquad
G/[G,G]\cong\mathbb C^\times,\qquad
\pi(f_{a,b})=a.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 149 / 3 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-149/3/a/solution